Claes Thornberg wrote:
> Why is it that you (on FIBS) gain more from a 2ptr than from a 1ptr,
> when we all know that in a perfect world a 2ptr is not different from
> a 1ptr? (see kit woolsey monograph 'How to play tournament backgammon')
> Can we design a ratingsystem that works for all different kinds of
> matchlengths? Or is it better to have different ratings for different
> match lengths, and only allow matches of certain lengths?
I've looked into this question a little bit. One idea I
had was to use a different way of measuring the opportunity
for skill in the match. FIBS just uses the match length.
I figure a better indication of the amount of work involved
is the total number of dice rolls in the match. Or, even
better, the number of dice rolls in "contact" positions.
So how many "contact position rolls" are there in typical
matches of various lengths? To answer this, I scanned
through the Big_Brother database and counted up the rolls.
Here are the results.
Average Rolls Per Match in Big_Brother Matches
rolls matches
----- -------
Match length 1: 17329 / 359 = 48.27
Match length 3: 54800 / 639 = 85.76
Match length 5: 116451 / 907 = 128.39
Match length 7: 86353 / 498 = 173.40
Match length 9: 9992 / 44 = 227.09
Match length 11: 8230 / 31 = 265.48
Note that there are less than twice as many rolls in a
3-point match as there are in a 1-point match. So, if
counting rolls is a good indicator, it suggests that
opportunity for skill is not really proportional to the
length of the match.
The other idea I had was to do exactly what Claes suggested
in another post: Compute a match equity table for players of
different ability. This method is not perfect -- the
computation assumes that both players always make efficient
doubles. This will produce some error, but I think the
results are still useful.
The following table is constructed assuming the stronger player
wins 51% of the time and that 25% of all wins are gammons.
Match Equity Table between Players of Unequal Ability
-1 -2 -3 -4 -5 -6 -7 -8 -9
----- ----- ----- ----- ----- ------ ----- ----- -----
-1: .5100 .6974 .7636 .8300 .8549 0.9011 .9186 .9431 .9528
-2: .3226 .5100 .6067 .6770 .7496 0.8061 .8461 .8788 .9045
-3: .2563 .4158 .5124 .5849 .6602 0.7237 .7717 .8129 .8474
-4: .1871 .3441 .4394 .5123 .5883 0.6543 .7077 .7529 .7940
-5: .1611 .2717 .3651 .4381 .5144 0.5816 .6390 .6890 .7351
-6: .1121 .2128 .3003 .3717 .4475 0.5150 .5754 .6280 .6779
-7: .0934 .1715 .2513 .3179 .3903 0.4555 .5162 .5700 .6220
-8: .0664 .1367 .2082 .2714 .3393 0.4026 .4625 .5167 .5694
-9: .0558 .1093 .1722 .2292 .2925 0.3525 .4108 .4647 .5177
If this table is accurate, then a player who beats his opponent
51% of the time in a 1-point match can expect to win 51.77%
of the time in a 9-point match.
There are some interesting features in this table. It says a
2-point match gives the stronger player no more advantage than
a 1-point match. This is true if the weaker player always
doubles at his first opportunity. It also says a 4-point match
is no better for the stronger player than a 3-point match.
This may be an exaggeration (coming from the assumption of both
players making efficient doubles), but it is something to think
about.
To summarize, let me compare the relative skill levels
estimated by each of these three methods:
1. The FIBS method.
2. The counting dice rolls method.
3. The match equity table method.
We'll start by assuming that a 1-point match gives the
stronger player 1.00 units of advantage. Here are the
relative advantages predicted by each method for longer
length matches:
FIBS Rolls MatEq
---- ----- -----
Match length 1: 1.00 1.00 1.00
Match length 2: 1.41 1.00
Match length 3: 1.73 1.33 1.24
Match length 4: 2.00 1.23
Match length 5: 2.23 1.63 1.44
Match length 6: 2.45 1.50
Match length 7: 2.64 1.89 1.62
Match length 8: 2.83 1.67
Match length 9: 3.00 2.16 1.77
Match length 10: 3.16 1.84
Match length 11: 3.31 2.34 1.92
Both the counting-rolls and match-equity-table methods
give less credit to longer length matches than FIBS does.
Or, another way of saying this, they give more credit to
1-point matches.
The question of how much greater advantage a stronger
player has at longer length matches is an interesting one.
I'd like to hear other people's ideas on this.
Tom